The moduli space of S¹-type zero loci for Z/2-harmonic spinors in dimension 3

Let M be a compact oriented 3-dimensional smooth manifold. In this paper, we construct a moduli space consisting of pairs (Σ, ψ) where Σ is a C¹-embedding simple closed curve in M, ψ is a Z/2harmonic spinor vanishing only on Σ, and ∥ψ∥_L2₁ ≠ 0. We prove that when Σ is C², a neighborhood of (Σ, ψ) in the moduli space can be parametrized by the space of Riemannian metrics on M locally as the kernel of a Fredholm operator.